Talk:Ordinal notation
Anyone know why the definition of \(C(\alpha, \beta)\) starts with \(n = 1\) and not 0? FB100Z • talk • 09:32, December 8, 2013 (UTC) That doesn't change the definition, as \(C_0(\alpha, \beta)\) is subset of \(C_1(\alpha, \beta)\), but I think definition will look nicier if we take union over all these sets. LittlePeng9 (talk) 21:31, December 8, 2013 (UTC) :But \(\vartheta(1) = \zeta_0\) Wythagoras (talk) 06:15, March 27, 2014 (UTC) has anyone managed to locate the source for weiermann's \(\vartheta\) in the final section? it's vel 01:34, September 22, 2014 (UTC) note to self: http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9096816 it's vel 01:46, September 22, 2014 (UTC) The version of Taranovsky's C that is shown in the page is not the strong one. The strong version is defined on comparisons. {hyp/^,cos} (talk) 09:55, October 9, 2014 (UTC) Why Feferman's \(\theta\) function has limit ordinal "N/A"? {hyp/^,cos} (talk) 10:00, December 3, 2014 (UTC) :Because I was not able to find its limit in my sources. I doubt that it is an unsolved problem, just one with an answer I haven't found yet. it's vel 15:26, December 3, 2014 (UTC) Extended veblen function Okay so the paper is here and I'm very confused on a few things... * what does rule C mean? i think it has to do with enumerating functions * why are people saying \(\phi(1, 0, 0) = \Gamma_0\) when zeros don't even seem to be allowed? * why is the \(\phi_\alpha(\beta)\) called the Veblen function when it doesn't look at all like what we see here? it's vel 17:05, October 12, 2014 (UTC) It's kind of like the situation with the Ackermann function, whcih generally refers to a specific function that is different from Ackermann's original construction. Ackermann's original construction is more complicated than need be, and Veblen's original Veblen function seems rather poor, actually. I think it's better to start from 0, and to have it so that adding or removing entries of value 0 don't change the value of the function, which are properties of the modern construction. Deedlit11 (talk) 17:27, October 12, 2014 (UTC) Theta function It's actually defined here up to BHO. Ikosarakt1 (talk ^ ) 05:17, February 12, 2015 (UTC) Klammersymbole JACKPOT. I found Schütte's original paper. -- ve 15:10, March 10, 2015 (UTC) :Hm, looks interesting. I can help you with the translation, if you want, but not now. Wythagoras (talk) 19:41, March 10, 2015 (UTC) ::Page 20 of http://www.cs.man.ac.uk/~hsimmons/TEMP/OrdNotes.pdf has an english translation, but it might not be an exact description -- ve 23:33, March 10, 2015 (UTC) Stegert's notation What's the definition? And how strong is it? {hyp/^,cos} (talk) 23:53, May 3, 2015 (UTC) :¯\_(ツ)_/¯ -- ve 03:47, May 4, 2015 (UTC) :Where have you heard of it? LittlePeng9 (talk) 04:17, May 4, 2015 (UTC) ::Stegert's dissertation, which is way too advanced for me to work through. -- ve 05:29, May 4, 2015 (UTC) ::Oh, it's too difficult for me to understand the dissertation and find the definition and strength. {hyp/^,cos} (talk) 13:02, May 4, 2015 (UTC) :::I can work through it but it'll take a lot of time and patience -- ve 01:12, May 5, 2015 (UTC) Confusing This is sooo confusing: for example, look at here: http://googology.wikia.com/wiki/Large_Veblen_ordinal Wich "Psi" function? There is lot of them! Most of articles on huge ordinals doesn't precise wich notation is used, and that confuse me a lot Fluoroantimonic Acid (talk) 11:54, June 30, 2015 (UTC) : I'm not sure if there is a single user on the wiki who knows what psi and theta functions are used all over the articles... LittlePeng9 (talk) 12:11, June 30, 2015 (UTC) Rathjen's notation We'd need a ruleset for the Rathjen's psi collapsing functions right in this article. I don't know its definition well enough, could someone help? ;A; Fluoroantimonic Acid (talk) 09:21, October 20, 2015 (UTC) Out of curiosity, why is it easier to manage values in Rathjen's notation if it is defined with two collapsing functions instead of one? Deedlit11 (talk) 00:18, May 17, 2016 (UTC) Intuitively speaking, the notation works a lot 'smoother', as we don't need to simultaneously manage \(\psi_\pi(\alpha)\) for \(\alpha\geq\text M\), which then feeds back into the function and makes it more difficult to make exact relations. We could just restrict \(\psi_\pi\) to values \(<\text M\), as done in the original text, but this also makes the definition of \(\chi\) more opaque. In addition, it becomes harder to prove that the behavior of \(\chi\) goes long the hyper-inaccessible hierarchy with the addition of the \(\psi_\pi\) functions to the definition - I think it's a lot more intuitively understandable with \9\chi\), at the least, defined separately. Case in point: me and another wiki user were discussing the behavior of the ordinals leading up to the PTO of KPM in relation to Hollom's new notation, and they claimed that the behavior of the \(\chi\) function doesn't work like that of the \(\psi_\pi\) functions, as we both completely missed that the definition of \(\psi_\pi\) had a restriction on \(\chi_\mu\), which made us both (afaik) think that \(\psi_{\omega_1}(\chi_{\Gamma_{\text M+1}}(0))=\psi_{\omega_1}(\text M)\). Separating this into two collapsing functions, although not strictly necessary, removes much of this apparent opacity in the definitions. ~εmli 09:18, May 17, 2016 (UTC) Comparing Veblen and Feferman In Ferermen's \(\theta\), it is written that for countable arguments, \(\theta_\alpha(\beta) = \varphi_\alpha(\beta)\). Is it true for \(\alpha = \Gamma_1\)? Obviously \(\theta_{\Gamma_1}(0) = \Gamma_0\). However normally \(\varphi_{\Gamma_1}(0) = \Gamma_1\). Therefore I think it is different. I think that Veblen function can be written as \begin{eqnarray*} C_0(\alpha, \beta) &=& \beta \cup \{0\}\\ C_{n+1}(\alpha, \beta) &=& \{\gamma + \delta, \varphi_\xi(\eta) | \gamma, \delta, \eta \in C_n(\alpha, \beta); \xi < \alpha\} \\ C(\alpha, \beta) &=& \bigcup_{n < \omega} C_n (\alpha, \beta) \\ \varphi_\alpha(\beta) &=& \min\{\gamma | \gamma \not\in C(\alpha, \gamma) \wedge \forall \delta < \beta: \varphi_\alpha(\delta) < \gamma\} \\ \end{eqnarray*} Definition of the Ferman's \(\theta\) is different from the Veblen's function in 2 ways. # Including the uncountables # Imposing \(\xi \in C_n(\alpha, \beta)\) in the definition of \(C_{n+1}(\alpha, \beta)\) The first point is irrelevant in the countable arguments. The second point restricts the \(\theta\) function to grow no further than \(\Gamma_0\) below \(\Omega\), which is very important for making the collapsing mechanism to work. �� Fish fish fish ... �� 18:47, May 13, 2018 (UTC) : As no one opposes, I corrected the text. �� Fish fish fish ... �� 12:59, May 17, 2018 (UTC) Proposal to separate the article into two We have so many ordinal collapsing functions in the article, while there is no main article focussing on ordinal collapsing functions. I think that it is better to create a new article on ordinal collapsing functions, and move the related contents in the article of ordinal notations to the new article. If nobody disagrees with it, I will do it. Please give me opinions. p-adic 01:27, December 26, 2019 (UTC) :Yes, I think it's reasonable. Triakula (talk) 05:02, December 26, 2019 (UTC) :: Thank you. I will wait a week for other opinions. :: p-adic 05:53, December 26, 2019 (UTC) :: Now a week has passed. I will separare the article into two later. :: p-adic 07:15, January 4, 2020 (UTC)